This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Hey Igimoh, good to see you again. First, let's list all possible outcomes when a coin is tossed thrice. The sample space S is: S = \HHH, HHT, HTH, THH, HTT, THT, TTH, TTT\ The total number of outcomes is |S| = 8. Let A be the event that there are exactly two heads. A = \HHT, HTH, THH\ So, |A| = 3. i) Find the probability that there are two heads given that at least one is a tail. Step 1: Define the event "at least one is a tail". Let B be the event that at least one is a tail. This means any outcome except HHH. B = \HHT, HTH, THH, HTT, THT, TTH, TTT\ So, |B| = 7. Step 2: Find the intersection of events A and B. The outcomes that are in both A (exactly two heads) and B (at least one tail) are: A B = \HHT, HTH, THH\ So, |A B| = 3. Step 3: Calculate the conditional probability P(A|B). The formula for conditional probability is P(A|B) = (P(A B))/(P(B)). P(A B) = (|A B|)/(|S|) = (3)/(8) P(B) = (|B|)/(|S|) = (7)/(8) P(A|B) = (3)/(8)(7)/(8) = (3)/(7) The probability is (3)/(7). ii) Find the probability that there are two heads given that the first is a tail. Step 1: Define the event "the first is a tail". Let C be the event that the first toss is a tail. C = \THH, THT, TTH, TTT\ So, |C| = 4. Step 2: Find the intersection of events A and C. The outcomes that are in both A (exactly two heads) and C (first is a tail) are: A C = \THH\ So, |A C| = 1. Step 3: Calculate the conditional probability P(A|C). P(A C) = (|A C|)/(|S|) = (1)/(8) P(C) = (|C|)/(|S|) = (4)/(8) = (1)/(2) P(A|C) = (1)/(8)(4)/(8) = (1)/(4) The probability is (1)/(4). Drop the next question!